The (1 + 1)-EA in Dynamic Environments

We study the $(μ+1)$ EA on the Binary Value function BinVal. We show that it needs at most $O(μ\log μ\cdot n \log n)$ function evaluations to find the optimum when $μ= o(n/\log n)$. This substantially improves upon the recent upper bound of $O(μ^5 n \log(n/μ^4))$ by Krejca, Neumann and Witt. Our results hold for several mutation operators including standard bit mutation. In particular, our bound implies that the $(μ+1)$ EA is at most a factor $O(\log μ\cdot \log n)$ slower on BinVal than on OneMax.

For the (1+1) evolutionary algorithm with standard bit mutation, we study the sensitivity of the expected hitting time $H_p=\mathbb{E}_x T$ to the mutation rate. We first point out an easily overlooked formalization pitfall: the improvement event is not monotone in the mutation mask, so the unsigned (total-influence) form of the Margulis-Russo formula does not apply; the correct object is the signed endpoint difference. Second, we give an exact three-dimensional separation: two fitness functions share the entire one-step success-rate curve, yet their expected hitting times are two different exact rational numbers; hence one-step success-rate quantities do not determine the expected hitting time. Building on the runtime derivative $H'_p=(I-Q_p)^{-1}Q'_p H_p$, we construct computable double-residual sign certificates, prove a per-initial-state sign theorem on OneMax (for every non-optimal initial state, $\partial_c H<0$ on $0<c<1$, where $p=c/n$; at $c=1$ only the distance-one state is stationary), and extend the framework to non-lumpable positive linear families: an explicit non-lumpability witness, a block-interval double-residual certificate that covers all states without enumerating them, a uniform sign bound $\partial_c\mathbb{E}T\le -9n/16$ over the whole interval $c\in[1/4,1/2]$ for an explicit family at all even scales $n\ge 8$, and a heterogeneous instance certificate $H'_x\le -1/6$ on 57 of 63 states across $c=1$. All finite verifications use exact rational arithmetic. A bounded systematic literature search did not uncover this exact combination, although the underlying tools are well established; we therefore make no novelty claim beyond the stated combination.

Problems defined on binary decision spaces have been intensively studied in the theory of multi-objective evolutionary algorithms (MOEAs). In contrast, no mathematical runtime analyses exist so far for MOEAs dealing with decision variables that take a finite number $r > 2$ of values, despite the prevalence of such problems in practice. In this work, we begin to fill this research gap. We analyze how the classic SEMO algorithm with unit-strength local mutation computes the Pareto front of an $r$-valued counterpart of the classic \oneminmax benchmark. For the expected number of function evaluations until the Pareto front is covered by the population of this MOEA, we prove an upper bound of $O(n^2 r^2 \log n)$ and a near-tight lower bound of $Ω(n^2 r (r + \log n))$. We can close the small remaining gap between these two bounds by considering a variant of the algorithm that accepts only strictly better solutions; for this variant, we show an upper bound of $O(n^2 r (r + \log n))$, matching our lower bound (which also holds for this variant). Our results suggest that classic MOEAs encounter no significant additional difficulties when dealing with multi-valued decision variables. However, significantly more advanced tools may be required to obtain tight bounds for algorithms with more complex population dynamics.